ARCHIVES OF TRANSPORT ISSN (print): 0866-9546
Volume 52, Issue 4, 2019 e-ISSN (online): 2300-8830
DOI: 10.5604/01.3001.0014.0211
Article is available in open access and licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0)
MODELLING VEHICULAR BEHAVIOUR USING TRAJECTORY
DATA UNDER NON-LANE BASED HETEROGENEOUS TRAFFIC
CONDITIONS
Hari Krishna GADDAM1, K. Ramachandra RAO2 1 Velagapudi Ramakrishna Siddhartha Engineering College, Vijayawada, Andhra Pradesh, India
2 Indian Institute of Technology Delhi, Department of Civil Engineering, Hauz Khas, New Delhi, India
Abstract:
The present study aims to understand the interaction between different vehicle classes using various vehicle attributes and
thereby obtain useful parameters for modelling traffic flow under non-lane based heterogeneous traffic conditions. To
achieve this, a separate coordinate system has been developed to extract relevant data from vehicle trajectories. Statistical analysis results show that bi-modal and multi-modal distributions are accurate in representing vehicle lateral placement
behaviour. These distributions help in improving the accuracy of microscopic simulation models in predicting vehicle
lateral placement on carriageway. Vehicles off-centeredness behaviour with their leaders have significant impact on safe longitudinal headways which results in increasing vehicular density and capacity of roadway. Another interesting finding
is that frictional clearance distance between vehicles influence their passing speed. Analysis revealed that the passing
speeds of the fast moving vehicles such as cars are greatly affected by the presence of slow moving vehicles. However, slow moving vehicles does not reduce their speeds in the presence of fast moving vehicles. It is also found that gap sizes
accepted by different vehicle classes are distributed according to Weibull, lognormal and 3 parameter log logistic distributions. Based on empirical observations, the study proposed a modified lateral separation distance factor and
frictional resistance factor to model the non-lane heterogeneous traffic flow at macro level. It is anticipated that the
outcomes of this study would help in developing a new methodology for modelling non-lane based heterogeneous traffic.
Keywords: Non-lane discipline, heterogeneous traffic, vehicle trajectories, lateral separation distance, frictional clearance
To cite this article:
Gaddam, H. K., Rao, K.R., 2019. Modelling vehicular behaviour using trajectory data
under non-lane based heterogeneous traffic conditions. Archives of Transport, 52(4),
95-108. DOI: https://doi.org/10.5604/01.3001.0014.0211
Contact: 1) [email protected] [https://orcid.org/0000-0002-8878-713X], 2) [email protected] [https://orcid.org/0000-0002-
7229-519X]– corresponding author
96
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
1. Introduction
In developing countries such as India, vehicles do
not follow lane discipline and they always deviates
from center-line positions. In addition disruptive
lane changing can also be observed. In addition, the
complexity in traffic flow also increases due to het-
erogeneous vehicle-driver units and their complex
interactions. Contemporary macroscopic continuum
models (Aw and Rascle 2000; Zhang 2002; Jiang et
al. 2002; Wong and Wong 2002; Logghe and Im-
mers 2003; Chanut and Buisson 2003; Gupta and
Katiyar, 2006; Gupta and Katiyar, 2007; Tang et al.
2009; Ngoduy 2011) are developed to model lane
based traffic movement. These models have some
limitations to completely capture the complexities
arise due to non-lane based heterogeneous traffic
movement. Recently, Nair et al. (2011) proposed po-
rous flow approach to model the behaviour of mo-
torised two wheeler in heterogeneous traffic envi-
ronment using static speed – pore size density rela-
tionship. Collecting pore size distribution of vehi-
cles in dynamically changing environment is cum-
bersome and moreover it is difficult to implement
the model for system with more than two vehicle
classes. In another study, Mohan and Ramadurai
(2013) addressed two main behavioural aspects of
heterogeneous traffic: dissimilar vehicle types and
non-lane discipline using extended Aw-Rascale
(2000) model. They used Area-Occupancy parame-
ter instead of linear density parameter. However the
model does not consider the vehicles off-cen-
teredness and the frictional effects of slow moving
vehicles in the traffic stream. In another approach,
Gupta and Dhiman (2014) proposed a non-lane con-
tinuum model using lateral separation distance fac-
tor. However, the model can only describe the vehi-
cle movement on a single lane road and it does not
taken in to account the interaction between slow
moving and fast moving vehicles, viscosity effects.
To this end, it is necessary to develop a second order
macroscopic model which explicitly describes the
heterogeneous traffic movement in non-lane based
traffic system in multi-lane environment. One of the
objectives of this study is to empirically derive some
information (suitable parameters) from vehicle tra-
jectories to build such models.
Several studies have been conducted to determine
the characteristics of non-lane based heterogeneous
traffic using empirical data. According to Dey et al.
(2006) study, speed distribution curves may be uni-
modal or bi-modal based on the speed variation
amongst different class of vehicles. This study re-
vealed that the proportion of slow moving vehicles
is not a true representative factor for bi-modality in
the speed data. In another study, Chunchu et al.
(2010) explored several traffic characteristics such
as the lateral placement of vehicles on the carriage-
way, lateral and longitudinal gaps between the vehi-
cles. They examined the relationship between lateral
gaps and area occupancy for various vehicle combi-
nations and they found consistent correlation be-
tween these two variables. Modelling headways in
non-lane based heterogeneous traffic condition is
critical and it is useful in developing simulation
models. Sharma et al. (2011) have done the explor-
atory data analysis to capture various parameters on
two-way undivided roadways. Traffic flow charac-
teristics such as arrival headways and speed distri-
butions have been studied in a systematic way.
Dubey et al. (2012) proposed Generalized Pareto
(GP) and Generalized Extreme Value (GEV) distri-
butions to model time gaps over a wide range of
flows from 550 veh/h to 4100 veh/h. These models
are successful in considering the problem of simul-
taneous arrival of vehicles in wide roads. In another
approach, Ambarwati et al. (2014) developed a class
specific pore size –density distribution, class spe-
cific speed-density and flow-density diagrams using
trajectory data in Surabaya city, Indonesia. The anal-
ysis revealed that motor cycles and other vehicles
exhibit significant difference in critical pore size dis-
tribution and traffic flow relationships. Kanagaraj et
al. (2015) investigated some of the microscopic flow
characteristics such as speed, acceleration and decel-
eration, selection of lateral spacing and longitudinal
distances of various vehicle classes for the data col-
lected on urban roads located in Chennai, India. The
results are found to be useful in development of
driver behaviour models for heterogeneous traffic
conditions. Dehghani and Tafti (2018) studied the
effect of different factors such as driver behaviour,
vehicle characteristics and environmental conditions
on saturation flow rates and capacity of signalised
intersection under weak lane heterogeneous traffic
conditions in Iran. The objective of this study is to
identify the suitable method to estimate the satura-
tion flow rates at the signalised intersections by
comparing empirical observations and estimated
values from different analytical models. In few other
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
97
studies such as Koshy and Arasan (2005), Dey et
al.(2008) and Asaithambi et al. (2012), several prob-
lems of non-lane heterogeneous traffic for instances
influence of composition, variability in physical and
dynamical characteristics and the presence of bus
stops on the capacity of the road are studied using
simulation models. The desired microscopic data to
develop these models such as speed, placement, ar-
rival and overtaking is obtained through field stud-
ies. To this end, it is understood that use of the pa-
rameters developed from the empirical data have rel-
evance in developing simulation models and model-
ling traffic flow characteristics.
Due to the complexity of Non-Lane based Heteroge-
neous Traffic system (NLHT), a detailed examina-
tion of vehicle interaction is required. Further it is
necessary to identify suitable parameters to build
macroscopic continuum models for non-lane sys-
tem. In this paper, an exploratory and confirmatory
data analysis is performed to examine the vehicle
trajectory data. Important vehicle characteristics
such as lateral placement on carriage way, effective
gap size distribution, relationship between longitu-
dinal headways and lateral separation distance,
moreover the dependency between vehicle passing
speed and lateral clearance of vehicles are studied.
Further, based on the observations, the study also in-
troduced new macroscopic parameters to model
non-lane systems using continuum theories. This
study helps in understanding the vehicle interactions
in mixed traffic and can be used for developing new
macroscopic continuum methodology for modelling
heterogeneous traffic in non-lane based systems.
2. Definition of heterogeneity and vehicle at-
tributes in non-lane based system
Vehicle behaviour in non-lane based heterogeneous
traffic stream significantly deviates from homogene-
ous traffic stream. The typical behaviour of vehicles
in NLHT can be best explained by staggered vehicle
movement, lane sharing, varying physical dimen-
sions and diverse dynamical characteristics. Due to
their distinct behaviour, they may increase or de-
crease the capacity of the traffic facility. One of the
unique features of the NLHT stream is that they uti-
lise the road width very effectively without compro-
mising their desired speed (Khan and Maini, 1999;
Mallikarjuna and Rao, 2006).
In this study, exploratory data analysis was done to
understand the behaviour of heterogeneous traffic
using the following attributes:
(i) Selection of lateral lane position (LP) by dif-
ferent vehicle classes across the carriage way
(Fig. 1(a)).
(ii) Relationship between longitudinal headways
(LH) and lateral separation distances (LSD)
(Fig. 1(b), 1(c)).
(iii) Relationship between vehicle passing speed
and lateral clearance (Fig. 1(d)).
(iv) Finally, the distributions of effective roadway
width (sum of vehicle width and frictional
clearance on both sides) required for the vehi-
cle classes to move downstream (Fig. 1(e)).
The coordinate system and the method of data col-
lection used in this study are shown at the bottom of
each sub plot in Fig.1. The analysis has been done
for the data collected at or near-capacity condition
where vehicles start interacting each other and suffi-
cient deviation in speeds can also be observed. Ter-
minologies such as non-lane based traffic and mixed
traffic are interchangeably used in this paper to rep-
resent the traffic streams in NLHT.
3. Location details and data collection
A straight section (100 m length × 10.5 m width) on
an urban arterial (Fig. 2) without any interruption
from bus bays and roadside facilities was chosen to
collect the traffic data and the section is located on
Panchsheel Marg, Outer Ring Road, Delhi, India.
Two hours video graphic survey was conducted to
obtain traffic and vehicular characteristics. In order
to obtain the data, camera was mounted on a foot
over bridge at 45𝑂 to 60𝑂 angle and at a height of
10 meters. Vehicle trajectories were obtained using
MATLAB® based video image processing tool de-
veloped in Traffic and Transportation Laboratory at
IIT Delhi for which proper calibration and validation
has been done before using the tool (Singh et al.,
2016). A separate coordinate system (as shown in
Fig. 1) is used to identify the vehicle position in each
time step, i.e. one second. In this study, vehicles are
grouped into four distinct types such as Cars, Motor-
ised Two Wheelers (MTW), Motorised Three
Wheelers (MThW) and Heavy Vehicles (HV) based
on their physical and dynamic characteristics. Phys-
ical and dynamical characteristics of different vehi-
cle classes are mentioned in Table 1 and the compo-
sition of these vehicle classes are given in Fig. 2.
98
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
Fig. 1. Studying heterogeneous vehicle behaviour under non-lane discipline (a) lateral placement (b) longitu-
dinal headway in meters (c) lateral separation distance and (d) a = frictional clearance between car and
three wheeler, b = frictional clearance between car and median (e) Effective carriageway width. Here
w = width of the vehicle, x = longitudinal co-ordinate, y = lateral co-ordinate.
Table 1. Vehicle classes and dimensions Vehicle
Class
Vehicles
included
Vehicle ave-
rage dimen-sions (m)
Speed characteri-
stics* (km/h)
vfree vcong vmean vσ
Car Small Car,
SUV*, Van
5.0 x 2.0 73.4 4.7 47 14.8
Motorised
Two Whe-eler
Scooter,
Moped
1.8 x 0.6 65.2 7.4 46.5 12.8
Motorised
Three
Wheeler
Auto –
Rickshaw,
Tuk-Tuk, LCV*
2.6 x 1.4 55.5 4.5 31 8.2
Heavy ve-
hicles
Bus, Truck 10.3 x 2.5 52.3 3.5 29 9.0
*SUV=sports utility vehicle, LCV = light commercial
vehicle, vfree=maximum free flow speed , vcong=minimum
congested speed, vmean=mean speed, vσ= standard devia-tion of speeds
Empirical traffic data and fundamental diagrams de-
rived from the trajectories have been used to check
the temporal variation of traffic flow, composition
of traffic, and free flow or congested conditions.
Trajectory data was used to obtain vehicle behav-
ioural attributes such as lateral placement, longitudi-
nal headways, lateral separation distances and effec-
tive lane width etc.
4. Data Analysis
The data extracted from vehicle trajectories pertain-
ing to different vehicle characteristics (as discussed
under section 2.0) are analysed using several statis-
tical methods. The analysis was done using statisti-
cal packages R (R Core Team, 2015) and Minitab
(Minitab, 2003).
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
99
Fig. 2. Location details and composition of vehicles
4.1. Lateral placement of vehicles on carriage-
way
To understand the driver’s choice in selecting the
lateral positon on carriageway, data has been col-
lected regarding lateral placement of vehicles, by
measuring the distance of right wheel from the me-
dian of the carriageway. Vehicles lateral placement
data for the Cars and MTW’s has multiple peaks and
it is only fitted by multi-modal distributions such as
normal, lognormal, and gamma mixture distribu-
tions (Fig. 3, Table 2). The distributions are fitted
using “mixdist” package in R (Macdonald and Du,
2011).The distributions and its goodness-of-fit
measures are presented in Table 2. Following infe-
rences can be drawn from the analysis.
− Vehicle lateral placement data (Table 2 and Fig. 3)
in mixed traffic conditions revealed that vehicles
do not restrict their movement to the center of the
lanes and they distribute across the carriageway.
Table 2. Statistical description about vehicles lateral placement and fitted distributions Vehicle
type
Mean ±
SD*
(m)
Median
(m)
Min*
(m)
Max*
(m)
Inference:
ANOVA*
statistics
Games-Howell
Multiple comparison
Distribution fitted
(chi-square value)
Car 3.74 ± 2.44 3.24 0.68 10.76 p < 0.00 Except MTW-HV
(p -value = 0.98), MThW-HV
(p -value = 0.19) all other are significantly different in mean
lateral positions
Lognormal Mixture (0.15)
MTW 6.11 ± 2.29 6.04 1.11 10.95 Gamma Mixture (0.04)
MThW 5.35 ± 1.91 5.03 1.44 9.87 Gamma (0.70)
HV 6.17 ± 1.51 6.05 4.00 8.03 Lognormal (0.92)
ALL 4.95 ± 2.52 4.86 0.69 10.95 Lognormal Mixture (0.01)
*SD = standard Deviation, Min = minimum, Max = maximum, ANOVA = analysis of variance
N
28°32'35.5"N 77°12'44.7"E
Panchsheel Marg Vehicle composition
100
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
Fig. 3. Lateral distribution of vehicles across the carriage way under non-lane discipline
(a) Total vehicles (b) Car (c) MTW (d) MThW (e) HV (Red triangle shows the mean value)
− The statistical analysis (Table 2 and Fig. 4) shows
that vehicles are choosing their respective posi-
tions based on their physical and dynamical char-
acteristics, ease of movement etc. Heavy vehicles
mostly occupy left most part of the carriageway to
give way to the fast moving vehicles whereas cars
mostly travel in the right most lanes to avoid hin-
drance from slow moving vehicles. On the other
hand, due to the smaller cross sectional area and
high maneuverability, MTW’s are able to occupy
anywhere on the carriageway and high percentage
of MThW chose to travel at middle of the carriage-
way.
− Vehicle lateral placement data seems to follow
multi-modal distributions. These distributions
help in improving the accuracy of microscopic
simulation models in predicting vehicle lateral
placement on carriageway.
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
101
(a) (b)
Fig. 4. (a) Box plot (b) Games –Howell comparison of mean lateral positioning of vehicles (In Figure 4(b)
Blue line represents that, if the interval does not contain zero, the corresponding means are signifi-
cantly different)
4.2. Lateral separation distance and longitudinal
headways for different vehicle classes
In heterogeneous traffic conditions, vehicles deviate
from their center-line position as shown in Fig. 1(c).
It happens due to the presence of different vehicle
sizes and the driver behaviour. Due to the off-cen-
teredness of the vehicles, following vehicle does not
assign full leadership to the front vehicle. In this
case, the selection of safe headway by any vehicle
depends on the amount of vehicles off-centeredness
and the type of vehicle present ahead. This charac-
teristic influences the number of vehicles present in
a roadway section and the highway capacity. To this
end, this section explores the relationship between
longitudinal headways and lateral separation dis-
tance between different classes of vehicles. The sec-
tion also presents lateral separation distance param-
eter estimation procedure and its usefulness in mac-
roscopic modelling methodology.
Table 3 shows that there is a large variation amongst
the vehicle classes in selecting longitudinal
headways and lateral seperation distances while
following the leader vehicle. It was observed that
vehicles maintained larger headways while
followoing heavy vehicles and those values are
ranging from 20 m to 52 m. Maximum LSD values
observed while vehicles following heavy vehicles
are from 1.6 m to 4.2 m. In contrast, vehicles
maintain smaller headways and lateral seperation
distances with MTW’s. For example, in case of Car-
MTW combination, LH was 9.30 m and LSD was
0.77 m. Further, ANOVA statistics (Table 3) also
prove that at peak traffic flow condition, mean lon-
gitudinal distance and mean lateral separation dis-
tances maintained by different follower and leader
combinations are significantly different (p < 0.00).
However, the Tukey - pairwise comparison shows
that vehicles maintained larger headways and larger
lateral separation distances with slow moving vehi-
cles such as MThW’s and heavy vehicles. From the
analysis, it can be inferred that presence of slow
moving vehicles reduce the density and capacity of
the traffic stream. Following interpretations can be
drawn from the scatter plots (Fig. 5) and from Tab.3.
It can be seen that the spatial headways are highly
correlated (-ve) with the lateral separation distances.
Further, it is also observed that the critical headways
decreases with increasing lateral separation between
the vehicles. Longitudinal headways of the follow-
ing vehicles do not get affected by the presence of
MTW as a leader. Following vehicles such as cars
and MThW maintain close proximity with MTW at
all conditions. Moreover, MTW drivers maintain
close distances to other vehicles and tend to overtake
whenever sufficient gaps are available. Maximum
longitudinal headways maintained by any class of
vehicle with MTW at zero LSD is 12 m and it indi-
cates that, the presence of MTW increases the ca-
pacity of the stream. Analysis also shows that heavy
vehicles barely follow any other vehicle in mixed
traffic stream. They create vacuum in front due to
their slow acceleration characteristics and it further
reduces the capacity.
HVMThWMTWCar
12
10
8
6
4
2
0
Late
ral D
ista
nce
(m
)
Vehicle Class
102
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
Table 3. Statistics analysis between different vehicle groups and lateral separation distance factor
Fo
llo
wer
-
lead
er v
ehic
le
typ
es
LH Description LSD Description
Correlation be-tween LSD and
LH
LSD fac-
tor for vehicle
combina-
tions (δij)
LSD factor
for each ve-
hicle
class (𝜹𝒊)
μ*
(S
D*
)
med
ian
min
*
max
*
AN
OV
A*
μ (
SD
)
med
ian
min
max
AN
OV
A*
Car-Car 17.84(5.95) 16.63 8.05 35.35
Wel
ch t
est
- p
< 0
.00
0.84(0.65) 0.71 0.03 2.40
Wel
ch t
est
- p
< 0
.00
y = -6.1x + 23
R² = 0.44 0.24
0.25
Car-MTW 9.62(4.12) 9.30 0.39 21.88 0.77(0.55) 0.68 0.01 2.12 y = -3.3x + 12
R² = 0.20 0.22
Car-
MThW 11.14(4.85) 10.89 0.74 23.78 1.17(0.73) 1.19 0.03 2.99
y = -4.3x + 16
R² = 0.43 0.33
Car-HV 25.52(12.97) 26.43 3.4 38.69 1.99(1.69) 1.57 0.50 4.17 y = -6.2x + 38
R² = 0.7 0.57
MTW-Car 17.19(5.92) 16.85 5.89 32.55 0.99(0.52) 0.96 0.03 2.01 y = -5.8x + 23
R² = 0.26 0.28
0.24
MTW-MTW
7.65(3.50) 7.33 0.41 20.53 0.64(0.39) 0.59 0.01 1.56 y = -3.5x + 10
R² = 0.15 0.18
MTW-
MThW 10.08(5.09) 10.67 0.26 23.5 1.04(0.55) 1.02 0.03 2.46
y = -6.3x + 17
R² = 0.46 0.30
MTW-HV 25.62(14.89) 25.40 3.62 52.12 1.80(0.77) 1.85 0.36 3.53 y = -14.7x + 52
R² = 0.58 0.51
MThW-
Car 15.05(5.09) 15.90 6.07 25.22 1.22(0.73) 1.31 0.10 2.38
y = -5.6x + 22
R² = 0.64 0.35
0.30
MThW-MTW
8.06(3.49) 7.85 2.62 18.04 0.66(0.51) 0.55 0.01 1.76 y = -4.1x + 11
R² = 0.36 0.19
MThW-
MThW 11.02(4.99) 11.87 1.20 22.14 1.17(0.61) 1.27 0.04 2.14
y = -5.3x + 17
R² = 0.40 0.33
MThW-HV
28.70 (7.68) 28.95 13.4 30.51 2.58(0.95) 2.11 1.47 4.12 y = -4.2x + 40
R² = 0.30 0.74
HV-All 21.22(8.19) 20.79 9.76 40.5 0.92(0.71) 0.72 0.04 2.33 y# = -4.0 x + 25,
R² = 0.12 0.26 0.26
*μ = mean (m), SD = Standard Deviation(m), min = minimum(m), max = maximum(m), ANOVA = analysis of vari-ance, y = longitudinal Headway (LH), x = lateral separation distance (LSD), # with respect to all vehicles
Lateral separation distance factor (δ)
Recently, Jin et al. (2010) and Li et al. (2015) have
proposed two different non-lane based full velocity
difference (NLBCF) car following models by con-
sidering lateral separation distances for single lane
and multilane traffic flow facilities respectively.
These models assume that the following vehicle
movement is governed by the lateral separation ef-
fects of its leader. Lateral effects of lane width helps
in improving the stability of traffic flow and explains
the traffic congestion pattern and its evolutions.
Even though the findings of these studies provide
some insights in analysing performance of non-lane
traffic system they lack in empirical understanding
of heterogeneous vehicle behaviour in non-lane
based traffic conditions. Therefore, this study modi-
fied the estimation procedure for finding lane sepa-
ration distance factor for heterogeneous traffic. The
modified lane separation distance factors for hetero-
geneous traffic stream given in Table 3 are derived
using Eq. (1).
δij =LSDij
W and δi = ∑ Pijδij
Nj=1 (1)
where δij is the lateral separation distance factor be-
tween vehicle i and j, LSDij is the lane separation
distance (m) between vehicle i and j and ‘W’ in de-
nominator represents the standard lane width (for
example 3.5 m is the standard lane width for Indian
traffic condition). Pij represents the number of times
vehicle i follow vehicle j.
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
103
Fig. 5. Relationship between longitudinal headway and lateral separation distance for different types of ve-
hicle interactions
104
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
4.3. Vehicle passing speed vs lateral clearance
In this section, lateral clearance between the vehicles
and its effect on speeds are analyzed. Regression
analysis (Fig. 6 and Table 4) shows that speed of the
cars is significantly affected by lateral clearance be-
tween the vehicles (p-value of L is 0.00, R² = 0.63).
In contrast, the speeds of vehicles such as MTW,
MThW and HV are not influenced by the presence
of other vehicles (p-value of L > 0.2 with R² ˂ 0.05).
Analysis further revealed that passing speeds of the
fast moving vehicles such as cars are greatly affected
by the presence of slow moving vehicles. However,
slow moving vehicles such as HV’s do not reduce
their speeds in the presence of fast moving vehicles.
Vehicles such as MTW’s and MThW’s managed to
travel at their desired speeds because of their ability
to seep through small gaps in the stream. These in-
puts help in framing new methodology in modelling
traffic flow. The governing equation for modelling
dynamic behaviour of vehicles is presented in the
next section.
Fig. 6. Correlation between passing speed (V) of the vehicle and lateral clearance (L)
Table 4. Regression statistics Response variable Predictor variables Coefficients t-statistics p-value R2-value
Car speed Intercept 11.81 16.18 0.00
0.63 Lateral Clearance 4.40 6.14 0.00
MTW speed Intercept 15.68 12.86 0.00
0.04 Lateral Clearance 1.57 1.19 0.24
MThW speed Intercept 14.24 8.80 0.00
0.05 Lateral Clearance 1.78 1.31 0.21
HV speed Intercept 13.45 5.60 0.00
0.04 Lateral Clearance 1.28 0.74 0.47
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
105
Frictional clearance factor
In existing higher order continuum modelling meth-
odology (Aw and Rascle 2000; Zhang 2002; Jiang et
al. 2002; Gupta and Katiyar, 2006; Gupta and Kati-
yar, 2007) the longitudinal acceleration of the vehi-
cle is governed by relaxation term, anticipation term
and convection term in the equation. However, in
non-lane system, one additional term is also required
to capture the effect of frictional resistance offered
by sideways movement of vehicles. This section dis-
cuss the introduction of frictional resistance term in
to the macroscopic model, which is as follows:
The vehicle acceleration in non-lane heterogeneous
traffic stream is governed by spatial headway and
velocity difference between different vehicle clas-
ses. Therefore, acceleration of ith class vehicle
(Tang et al., 2009) is a function of:
dvi,n(t)
dt= 𝑓( Vi
n(𝑡), ∆Vin,n+1
(t), ∆xin,n+1(t))
(2)
dvi,n(t)
dt= ki[Vi,n(∆xi,n) − vi,n] +
∑ λij
N
j=1pij[vj,n+1 − vi,n]
(3)
∀j = 1, 2…N
where:
∆xin,n+1(t) = ∑ Pij
Nj=1 (xn+1
j (t) − xni (t)),
∆Vin,n+1(t) = ∑ Pij
Nj=1 (Vn+1
j (t) − Vni (t))
are space headway and speed of vehicle class i re-
spectively. N is the number of vehicle classes; Pij is
the number of times vehicle class i followed vehicle
class j; αi=1/Ti and κi= 1/τi are the driver reactive
coefficients of vehicle class i.
In order to develop macroscopic continuum model,
suitable transformation technique must be used to
convert discrete variables into the continuous varia-
bles. The method suggested by (Jiang et al., 2002)
is applied to transfer the variables from microscopic
to macroscopic ones. After applying Taylor expan-
sion series and neglecting the higher order terms, the
final form of the model is:
∂Vi
∂t+ Vi
∂Vi
∂x=
1
Ti
[Vie(𝑘) − Vi] +
∑Pij
τij∆x
∂Vj
∂x
N
j=1
∑Pij
τij
N
j=1
(Vj(x, t) − Vi(x, t))
(4)
The first term in the right hand side of the equation
represents the relaxation term, second term repre-
sents the driver reactions to sudden change in the
downstream velocity. Third term is the velocity dif-
ference between two different vehicle classes pre-
sent in the same cell.
Even though the macroscopic continuum model pre-
sent in Equation (4) is logically sound, some engi-
neering corrections need to be applied to capture
complex driving behaviour present in Indian driving
environment. As discussed in section 4.3, only cars
are significantly affected by the presence of other
vehicle classes present in the same section. Based on
the empirical observations, a new term called fric-
tion factor (𝜇𝑖𝑗) is introduced to modify the last term
in the equation.
μijaij where aij = ∑Pij
τij
N
j (≠i)
(Vj(x, t) − Vi(x, t)) (5)
where 𝜇𝑖𝑗 is the friction factor and the value is 1 if
Vif > Vjf otherwise zero. Pij is the percentage of times
vehicle i followed j, 𝜏𝑖𝑗is the reaction time and Vj
and Vi are speeds of vehicle j and i respectively. It is
expected that the proposed equation will improve
the model capability in capturing non-lane behav-
iour.
4.4. Effective gap size
Effective gap size is the minimum width of the road
required for the vehicle to move downstream with-
out reducing its current speed significantly. It is es-
timated to be the width plus frictional clearance on
both sides of the vehicle (Fig.1 (e)). Gap size ac-
cepted by different vehicle classes have been esti-
mated using the method suggested in section two. It
is found that, vehicles effective gap sizes are distrib-
uted according to Weibull, lognormal and 3 param-
eter log logistic distributions with mean effective
sizes ranging from 1.83 m to 4.99 m. The wide-
spread in effective gap size data of Cars and
MThW’s shows that drivers are selecting different
gap sizes. Interestingly, MTW and HV data is
skewed to the right and distributed around the mean
value. Descriptive statistics and gap size distribu-
tions are given in Table 5 and Fig. 7. These results
will be used in estimating traffic density and model-
ling vehicle behaviour.
106
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
Fig. 7. Gap size distribution for different vehicles classes
Table 5. Descriptive statistics and distribution of effective gap sizes
Vehicle type
(sample size)
Mean
(m)
Median
(m)
Minimum
(m)
Maximum
(m)
Standard
Deviation
(m)
Skewness Kurtosis Distribution
fitted
(p-value)
P-value
Car (70) 2.90 2.91 1.80 3.78 0.46 -0.44 -0.26 Weibull 0.24
MTW (143) 1.83 1.79 0.75 2.91 0.44 +0.11 -0.32 Log Normal 0.31
MThW (55) 2.52 2.57 1.69 3.32 0.38 -0.09 -0.75 Weibull 0.25
HV (20) 4.99 4.67 4.21 7.60 0.87 +2.32 +6.58 3 parameter
log logistic
0.02*
*Likelihood Ratio P-Value
Gaddam, H. K., Rao, K.R.,
Archives of Transport, 52(4), 95-108, 2019
107
5. Conclusions
Following are the conclusions from this study:
− In non-lane environment, vehicles lateral posi-
tions on carriageway depend on their ease of
movement, physical and dynamical characteris-
tics. The study suggests that the use of bi-modal
and multi-modal distributions in representing lat-
eral placement characteristics of vehicles will im-
prove the modelling accuracy.
− Safe longitudinal headways maintained by vehi-
cles decrease due to their off-centered behaviour.
This behaviour leads to reduction in the critical
gaps maintained by vehicles. In other words, it in-
creases the density of the road way section. Car
following models suggested by Jin et al. (2010)
and Li et al. (2015) can be taken as basis to in-
corporate off-centered behaviour of the vehicles
into macroscopic continuum model.
− Another interesting outcome of this study is that
frictional clearance distance between vehicles in-
fluence their passing speed. Based on empirical
observations, parameters such as lateral separa-
tion distance factor and frictional clearance factor
were introduced to study the behaviour of non-
lane heterogeneous traffic flow at macro level.
− It is interesting to note that vehicles maintain
closer headways with MTW’s. Hence, high pro-
portion of MTW’s increases the density and ca-
pacity of the traffic stream. However, heavy ve-
hicles such as buses act like a moving bottle-
necks, thereby reducing the critical density, jam
density and capacity of the traffic stream.
− The two new concepts proposed in this study such
as modified lateral separation distance factor and
frictional clearance factor can be used in the de-
velopment of non-lane based heterogeneous con-
tinuum models.
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